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 I. Basic math.
 II. Pricing and Hedging.
 1 Basics of derivative pricing I.
 A. Single step binary tree argument. Risk neutral probability. Delta hedging.
 B. Why Ito process?
 C. Existence of risk neutral measure via Girsanov's theorem.
 D. Self-financing strategy.
 E. Existence of risk neutral measure via backward Kolmogorov's equation. Delta hedging.
 F. Optimal utility function based interpretation of delta hedging.
 2 Change of numeraire.
 3 Basics of derivative pricing II.
 4 Market model.
 5 Currency Exchange.
 6 Credit risk.
 7 Incomplete markets.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Single step binary tree argument. Risk neutral probability. Delta hedging.

onsider a market consisting of a single stock and a money market account (MMA). The MMA is riskless and has rate . The price of the stock is given by a stochastic process . At present moment the stock is priced at . There is no trading or price evolution until a time moment , and at the stock may assume only two possible prices: and . Hence, we introduce the random events We would like to come up with a -price for a contract that pays at time a random amount

Suppose that at the moment we sell the contract for a price , purchase shares of the stock and put the remainder of the funds into the MMA. Any of the values may be negative. We assume that we can short the stock and borrow from MMA at the same rate .

The value of the position is zero: We hold the position until the time moment when there are two possibilities: Let us select and so that would be the same for both situations: We arrive to a system of two equations: The quantity is called "delta" and the strategy of taking position in the underlying stock equal to is called "delta hedging".

To determine we make the following "no arbitrage" argument. At we have a position of zero value. At the moment we have a position of set value (given our selection of ). Such set value has to be zero because otherwise we made or lost money in absence of risk, hence, the price of the contract would not be correct. Therefore, we have a third equation: from which we derive : and we obtain the correct price of the contract: We would like to put the above result into the form for some numbers . Hence, we rearrange the expression : thus Note that For this reason we call the numbers the "risk neutral probabilities": and represent the expression as the "risk neutral expectation of discounted payoff":

 Notation. Index. Contents.